An energy scale directly related to superconductivity in the high-T_c cuprate superconductors: Universality from the Fermi arc picture

We have performed a temperature dependent angle-resolved photoemission spectroscopy (ARPES) study of the tri-layer high-$T_c$ cuprate superconductor (HTSC) Bi$_2$Sr$_2$Ca$_2$Cu$_3$O$_{10+\delta}$ (Bi2223), and have shown that the \textquotedblleft effective\textquotedblright superconducting (SC) gap $\Delta_{\rm{sc}}$ defined at the end point of the Fermi arc and the $T_c$ (= 110 K) approximately satisfies the weak-coupling BCS-relationship 2$\Delta_{\rm{sc}}$ = 4.3$k_{\rm{B}}T_c$. Combining this result with previous ARPES results on single- and double-layer cuprates, we show that the relationship between 2$\Delta_{\rm{sc}}$ = 4.3$k_{\rm{B}}T_c$ holds for various HTSCs. Furthermore, at $T$ $\sim$ $T_c$, the quasi-patricle width at the end point of the Fermi arc is found to coincide with $\Delta_{\rm{sc}}$, consistent with the context of Planckian dissipation.